affine scheme

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In
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promine ...
, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together ...
s of ''R'', and is usually denoted by $\operatorname$; in algebraic geometry it is simultaneously a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
equipped with the sheaf of rings $\mathcal$.

# Zariski topology

For any
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
''I'' of ''R'', define $V_I$ to be the set of prime ideals containing ''I''. We can put a topology on $\operatorname\left(R\right)$ by defining the collection of closed sets to be :$\.$ This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of $\operatorname\left(R\right)$, and $\$ is a basis for the Zariski topology. $\operatorname\left(R\right)$ is a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a T1 space. However, $\operatorname\left(R\right)$ is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.

# Sheaves and schemes

Given the space $X = \operatorname\left(R\right)$ with the Zariski topology, the structure sheaf ''O''''X'' is defined on the distinguished open subsets ''D''''f'' by setting Γ(''D''''f'', ''O''''X'') = ''R''''f'', the localization of ''R'' by the powers of ''f''. It can be shown that this defines a
B-sheaf In mathematics, the gluing axiom is introduced to define what a sheaf \mathcal F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor ::(X) \rightarrow C to a category C which initial ...
and therefore that it defines a sheaf. In more detail, the distinguished open subsets are a basis of the Zariski topology, so for an arbitrary open set ''U'', written as the union of ''i''∈''I'', we set Γ(''U'',''O''''X'') = lim''i''∈''I'' ''R''''fi''. One may check that this presheaf is a sheaf, so $\operatorname\left(R\right)$ is a ringed space. Any ringed space isomorphic to one of this form is called an affine scheme. General schemes are obtained by gluing affine schemes together. Similarly, for a module ''M'' over the ring ''R'', we may define a sheaf $\tilde$ on $\operatorname\left(R\right)$. On the distinguished open subsets set Γ(''D''''f'', $\tilde$) = ''M''''f'', using the
localization of a module In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fraction ...
. As above, this construction extends to a presheaf on all open subsets of $\operatorname\left(R\right)$ and satisfies gluing axioms. A sheaf of this form is called a
quasicoherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
. If ''P'' is a point in $\operatorname\left(R\right)$, that is, a prime ideal, then the stalk of the structure sheaf at ''P'' equals the localization of ''R'' at the ideal ''P'', and this is a local ring. Consequently, $\operatorname\left(R\right)$ is a
locally ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf o ...
. If ''R'' is an integral domain, with field of fractions ''K'', then we can describe the ring Γ(''U'',''O''''X'') more concretely as follows. We say that an element ''f'' in ''K'' is regular at a point ''P'' in ''X'' if it can be represented as a fraction ''f'' = ''a''/''b'' with ''b'' not in ''P''. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe Γ(''U'',''O''''X'') as precisely the set of elements of ''K'' which are regular at every point ''P'' in ''U''.

# Functorial perspective

It is useful to use the language of category theory and observe that $\operatorname$ is a functor. Every ring homomorphism $f: R \to S$ induces a continuous map $\operatorname\left(f\right): \operatorname\left(S\right) \to \operatorname\left(R\right)$ (since the preimage of any prime ideal in $S$ is a prime ideal in $R$). In this way, $\operatorname$ can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime $\mathfrak$ the homomorphism $f$ descends to homomorphisms :$\mathcal_ \to \mathcal_\mathfrak$ of local rings. Thus $\operatorname$ even defines a contravariant functor from the category of commutative rings to the category of
locally ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf o ...
s. In fact it is the universal such functor hence can be used to define the functor $\operatorname$ up to natural isomorphism. The functor $\operatorname$ yields a contravariant equivalence between the
category of commutative rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings i ...
and the category of affine schemes; each of these categories is often thought of as the opposite category of the other.

# Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies ''algebraic sets'', i.e. subsets of ''K''''n'' (where ''K'' is an algebraically closed field) that are defined as the common zeros of a set of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
s in ''n'' variables. If ''A'' is such an algebraic set, one considers the commutative ring ''R'' of all polynomial functions ''A'' → ''K''. The ''maximal ideals'' of ''R'' correspond to the points of ''A'' (because ''K'' is algebraically closed), and the ''prime ideals'' of ''R'' correspond to the ''subvarieties'' of ''A'' (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets). The spectrum of ''R'' therefore consists of the points of ''A'' together with elements for all subvarieties of ''A''. The points of ''A'' are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of ''A'', i.e. the maximal ideals in ''R'', then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets). Specifically, the maximal ideals in ''R'', i.e. $\operatorname\left(R\right)$, together with the Zariski topology, is homeomorphic to ''A'' also with the Zariski topology. One can thus view the topological space $\operatorname\left(R\right)$ as an "enrichment" of the topological space ''A'' (with Zariski topology): for every subvariety of ''A'', one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the
generic point In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic ...
for the subvariety. Furthermore, the sheaf on $\operatorname\left(R\right)$ and the sheaf of polynomial functions on ''A'' are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemes.

# Examples

* The affine scheme $\operatorname\left(\mathbb\right)$ is the final object in the category of affine schemes since $\mathbb$ is the initial object in the category of commutative rings. * The affine scheme is scheme theoretic analogue of $\mathbb^n$. From the functor of points perspective, a point $\left(\alpha_1,\ldots,\alpha_n\right) \in \mathbb^n$ can be identified with the evaluation morphism . This fundamental observation allows us to give meaning to other affine schemes. * looks topologically like the transverse intersection of two complex planes at a point, although typically this is depicted as a $+$ since the only well defined morphisms to $\mathbb$ are the evaluation morphisms associated with the points $\$. * The prime spectrum of a Boolean ring (e.g., a power set ring) is a (Hausdorff)
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
. * (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) if and only if it is quasi-compact, quasi-separated and sober.

# Non-affine examples

Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together. * The Projective $n$-Space
Proj construction In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functo ...
(in fact, we can define Projective Space for any base scheme). The Projective $n$-Space for $n \geq 1$ is not affine as the global section of $\mathbb^n_k$ is $k$. * Affine plane minus the origin. Inside

# Non-Zariski topologies on a prime spectrum

Some authors (notably M. Hochster) consider topologies on prime spectra other than Zariski topology. First, there is the notion of constructible topology: given a ring ''A'', the subsets of $\operatorname\left(A\right)$ of the form $\varphi^*\left(\operatorname B\right), \varphi: A \to B$ satisfy the axioms for closed sets in a topological space. This topology on $\operatorname\left(A\right)$ is called the constructible topology. In , Hochster considers what he calls the patch topology on a prime spectrum.Willy Brandal, Commutative Rings whose Finitely Generated Modules Decompose By definition, the patch topology is the smallest topology in which the sets of the forms $V\left(I\right)$ and $\operatorname\left(A\right) - V\left(f\right)$ are closed.

# Global or relative Spec

There is a relative version of the functor $\operatorname$ called global $\operatorname$, or relative $\operatorname$. If $S$ is a scheme, then relative $\operatorname$ is denoted by $\underline_S$ or $\mathbf_S$. If $S$ is clear from the context, then relative Spec may be denoted by $\underline$ or $\mathbf$. For a scheme $S$ and a quasi-coherent sheaf of $\mathcal_S$-algebras $\mathcal$, there is a scheme $\underline_S\left(\mathcal\right)$ and a morphism $f : \underline_S\left(\mathcal\right) \to S$ such that for every open affine $U \subseteq S$, there is an isomorphism $f^\left(U\right) \cong \operatorname\left(\mathcal\left(U\right)\right)$, and such that for open affines $V \subseteq U$, the inclusion $f^\left(V\right) \to f^\left(U\right)$ is induced by the restriction map $\mathcal\left(U\right) \to \mathcal\left(V\right)$. That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the Spec of the sheaf. Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative $\mathcal_S$-algebras and schemes over $S$. In formulas, :$\operatorname_\left(\mathcal, \pi_*\mathcal_X\right) \cong \operatorname_\left(X, \mathbf\left(\mathcal\right)\right),$ where $\pi \colon X \to S$ is a morphism of schemes.

## Example of a relative Spec

The relative spec is the correct tool for parameterizing the family of lines through the origin of $\mathbb^2_\mathbb$ over $X = \mathbb^1_.$ Consider the sheaf of algebras and let $\mathcal = \left(ay-bx\right)$ be a sheaf of ideals of $\mathcal.$ Then the relative spec $\underline_X\left(\mathcal/\mathcal\right) \to \mathbb^1_$ parameterizes the desired family. In fact, the fiber over

# Representation theory perspective

From the perspective of representation theory, a prime ideal ''I'' corresponds to a module ''R''/''I'', and the spectrum of a ring corresponds to irreducible cyclic representations of ''R,'' while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is the study of modules over its group algebra. The connection to representation theory is clearer if one considers the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
, and writing in terms of $x_i$ corresponds to choosing a basis for the vector space. Then an ideal ''I,'' or equivalently a module $R/I,$ is a cyclic representation of ''R'' (cyclic meaning generated by 1 element as an ''R''-module; this generalizes 1-dimensional representations). In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in ''n''-space, by the nullstellensatz (the maximal ideal generated by $\left(x_1-a_1\right), \left(x_2-a_2\right),\ldots,\left(x_n-a_n\right)$ corresponds to the point $\left(a_1,\ldots,a_n\right)$). These representations of

# Functional analysis perspective

The term "spectrum" comes from the use in operator theory. Given a linear operator ''T'' on a finite-dimensional vector space ''V'', one can consider the vector space with operator as a module over the polynomial ring in one variable ''R''=''K'' 'T'' as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of ''K'' 'T''(as a ring) equals the spectrum of ''T'' (as an operator). Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module: : the 2×2 zero matrix has module : showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module : showing algebraic multiplicity 2 but geometric multiplicity 1. In more detail: * the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity; * the primary decomposition of the module corresponds to the unreduced points of the variety; * a diagonalizable (semisimple) operator corresponds to a reduced variety; * a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under ''T'' spans the space); * the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial.

# Generalizations

The spectrum can be generalized from rings to
C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuo ...
s in operator theory, yielding the notion of the
spectrum of a C*-algebra In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra ''A'', denoted ''Â'', is the set of unitary equivalence classes of irreducible *-representations of ''A''. A *-representation π of ''A'' on a Hilbert space ''H'' is irreduci ...
. Notably, for a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
, the algebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) is a ''commutative'' C*-algebra, with the space being recovered as a topological space from $\operatorname$ of the algebra of scalars, indeed functorially so; this is the content of the
Banach–Stone theorem In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to reco ...
. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to ''non''-commutative C*-algebras yields noncommutative topology.

*
Scheme (mathematics) In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different s ...
*
Projective scheme In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
*
Spectrum of a matrix In mathematics, the spectrum of a matrix is the set of its eigenvalues. More generally, if T\colon V \to V is a linear operator on any finite-dimensional vector space, its spectrum is the set of scalars \lambda such that T-\lambda I is not invert ...
* Serre's theorem on affineness * Étale spectrum * Ziegler spectrum *
Primitive spectrum In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals ...

* * * * * *